Problem 29 - General Chemistry: Principles and Modern Applications

Question

Three different sets of data of \([\mathrm{A}]\) versus time are giv the following table for the reaction \(A \longrightarrow\) prod [Hint: There are several ways of arriving at answer each of the following six questions. $$\begin{array}{cccccc} \hline \text { I } & & \text { II } & & \text { III } & \\ \hline \begin{array}{c} \text { Time, } \\ \text { s } \end{array} & \text { [A], M } & \begin{array}{c} \text { Time, } \\ \text { s } \end{array} & \text { [A], M } & \begin{array}{c} \text { Time, } \\ \text { s } \end{array} & \text { [A], M } \\ \hline 0 & 1.00 & 0 & 1.00 & 0 & 1.00 \\ 25 & 0.78 & 25 & 0.75 & 25 & 0.80 \\ 50 & 0.61 & 50 & 0.50 & 50 & 0.67 \\ 75 & 0.47 & 75 & 0.25 & 75 & 0.57 \\ 100 & 0.37 & 100 & 0.00 & 100 & 0.50 \\ 150 & 0.22 & & & 150 & 0.40 \\ 200 & 0.14 & & & 200 & 0.33 \\ 250 & 0.08 & & & 250 & 0.29 \\ \hline \end{array}$$ What is the approximate half-life of the first-order reaction?

Step-by-Step Solution

Answer

The approximate half-life of the first-order reaction is 75 seconds.

Step 1: Identify Presented Data

The concentration-time data is presented in three sets (I, II, and III). All sets start with the initial concentration of \([A]\) as 1.00 M. We are interested in Set I, as it corresponds to the first-order kinetic reaction in this exercise.

Step 2: Locate Half of Initial Concentration

Half of the initial concentration is 1.00 M / 2 = 0.50 M. In Set I, the given concentration closest to 0.5M is 0.47M. This concentration is observed at time 75 s.

Step 3: Determine the Half-Life

As the half-life is defined by the time it takes for concentration to decrease to half of its initial value, by inspection of the data, the half-life is approximately the same as the time when [A] became approximately 0.50M. Therefore, the half-life for this reaction is about 75 s.

Key Concepts

Chemical Kinetics

Chemical kinetics involves the study of how quickly chemical reactions occur. It provides insights into the speed of a reaction and the factors affecting this speed.
In kinetic studies, changes in the concentration of reactants and products over time are monitored to figure out the rate at which a reaction proceeds.

• Different reactions happen at different speeds. Some are incredibly fast, like explosions, while others are slow, like rusting.
• Understanding kinetics helps in optimizing conditions to speed up desirable reactions, such as in industrial manufacturing, or to slow down unwanted reactions, like food spoiling.

By examining the concentration-time data, as given in the table, we can determine the rate law of a reaction, which tells us how the concentrations of reactants influence the rate. It plays a crucial role in determining the half-life of reactions.

First-Order Reaction

In chemical kinetics, a first-order reaction has a rate that depends linearly on one concentration term.
In other words, the rate of reaction is directly proportional to the concentration of a single reactant.
If the concentration of the reactant decreases by half, the rate of the reaction also decreases by half.

• The rate law for a first-order reaction is written as: \[ ext{Rate} = k[A] \] where \( k \) is the rate constant, and \([A]\) is the concentration of the reactant.
• First-order reactions are characterized by having a constant half-life, irrespective of the initial concentration.

In the exercise, set I shows the concentration of \([ ext{A} ]\) decreasing gradually. By demonstrating that the half-life remains consistent, we can verify that the reaction is first-order.

Concentration-Time Data

Concentration-time data is crucial in kinetics as it records how the concentration of reactants changes over time.
Looking at this data helps to decide the reaction order and the reaction rate law.
For the given exercise, the data sets expose how the concentration of substance \([ ext{A} ]\) changes at certain time intervals.

• To determine the half-life, which is the time it takes for the concentration to decrease to half its original value, we identify the time at which the concentration reaches half of its initial value.
• In set I from the exercise, the concentration data reveals that \([ ext{A} ]\) falls from 1.00M to approximately 0.50M around 75 seconds, indicating the half-life.

Examining concentration-time data not only helps in calculating half-lives but also in comprehending the overall mechanism of the reaction.